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I tend to favor a more rulings over rules style of play, where nothing not expressly prohibited can be attempted. And to not get hung up on any particular rule but feel free to improvise. I've found that since 5e really supports that, most of the players fall in line with my style of preferences and we get along like peas in a pod.
There are far too many things I like about 5e to list them all here, but the biggest ones are:
Bounded Accuracy
Advantage/Disadvantage
Backgrounds
Proficiency Bonus
Subclasses
The Magic System (preparation separated from slots, at-will cantrips, rituals, concentration, etc.)
Magic Items being optional and precious
* Lair Actions
* Backgrounds generally and their decoupling from class specifically
* Inspiration with Bonds properly resolved or Flaws that drive legitimate, interesting complications
* Bounded accuracy w/ regards to : keeping low-level foes pertinent at higher levels
... I am so evil that I will not even name the source(s)! Ha!
This is a great point. The way backgrounds are done, as well as feats (I guess partly because they are optional in the first place), encourages you to custom make your own. Definitely one of my favourite things about 5e and an approach I feel was less prevalent in 3e and 4e.
Also I have to mention the Legendary template. I hate the name - just call them solos! - but love the mechanics. You really can threaten the party with a single beastie now. We do use a variant of Legendary Resistance allowing a save when normally there isnt one (like against Sleep).
I started a campaign for my two experienced older brothers and their inexperienced children, and they had a blast with the random backgrounds, which initially I thought was a waste of time but it added so much to the role playing aspect.
Hands down, speed of play. Holy CRAP does this game fly. I'd really forgotten how fast D&D used to play to be honest. It's been a lot of years since I played 2e, and, tbh, even in 2e you could grind combat pretty slow, depending on how many splats you were using.
But 5e? WOW. This thing is faster than ... a really fast thing through a goose. Fantastic.
Also liking the new XP tables where you skip pretty quickly through the first two or three levels and then slowly slow down at higher levels.
General:
Simplicity (easy to play, easy to understand, easy to run)
Flexibility (easy to modify/houserule...even on the fly)
Enabling of multiple play styles and stories (Finally! A D&D for Everybody!)
Fast Play (Players don't get Bored!)
Easy scaling of granularity during gameplay (from almost no-rules improv to very specific mechanics)